Re: Don't drive and drone

Hi Dan,

> If I've been working too intensely on micro-music-math type problems
> or scores I can get a very nasty "white line fever" type reaction when
> I finally haul myself somewhere and try to and sleep.

> Sometimes these are acutely unpleasant -- kind of like a vicious
> dizzying loop. But I swear on occasion I've seen things a thousand
> times more complicated that I can cogitate expanding and contracting
> right before me just beyond the threshold of retention.

> Like I wasn't exhausted enough already! That's kind of like talking in
> your sleep and saying "I'm so tired" -- madness!

Yes, I can't remember what particularly I was working on at the time,
but I know one of the things I was trying to understand was
Godel's proof of his incompleteness theorem for arithmetic.

That kind of thing can easily set one into a kind of mind boggling
infinite recursion. Actually you can even get a bit dizzy if one
focusses really hard on it.

Other times it is geometrical things that attract ones attention
in this same compelling way, e.g. trying to figure out all the faces
of some complex geometrical figure and how they fit together
- not that well advised a thing to do while cycling!

Other times, can be abstract algebraic things, or a particular
step in a proof that one is trying to figure out, or a key concept
one needs to define in order to carry through a proof, and one
kind of keeps worrying at it like a dog with a bone, trying
to figure it out. Then one might suddenly see it, and it may
be something really easy actually, at that point.

I well remember getting involved in all of those at one time or
another.

Yes, and sleep is what often gives it a chance to sort itself
out, it is fairly common I think for a mathematician to go to bed
with a proof in a very incomplete state, and wake up with the
key idea, or a few key ideas, so that it is all finished bar
the writing out and polishing up of the presentation.

I'm not sure whether that happens in ones sleep, or just very fast
in the relaxed moments while one awakens. At any rate, something
goes on there that seems to be sometimes way beyond ones mormal
problem solving ability - in terms of the speed with which it happens
anyway.

I'm inclined to think it actually happens very fast at the moment
one wakes up.

Can also happen in other situations. A famous mathematician
at Oxford got the key idea that was the basis for one of the
theorems he is famous for while crossing a street with someone
else. He just had a feeling of having glimpsed something
very significant, but couldn't remember what it was.
Then later, when on his own, he tried to remember
what it was, and eventually succeeded in recovering it.

That's Professor Roger Penrose and his theorem that
collapsing black holes will always collapse to a singularity and
can't avoid it by kind of collapsing through themselves
in an asymmetrical way and bouncing out again, no matter
what the initial state. Accounts of his experience in books.

The key ideas may be ones that can be expressed quite simply,
which is why one can remember them. That happens in maths -
you get an idea that is in essence very simple, you can
just see it, quite intuitively. But when you try to write it
down, it may take many words to follow it all through, with
all the steps in the proof just falling out in a perfect fashion.

One wonders then how it is that one got at them, how
one knew, or at least felt so sure, that this approach
was going to work.

All htis happens in a dramatic way for some, as in Professor
Penrose's experience. But it also happens in a more plodding
kind of way. Either when trying to prove something, or
when trying to understand a proof working through line
by line.

You worry away at it, and then often at a moment of relaxation
when no longer thinking about it, one then suddenly sees
what it was about.

Or I think in even smaller ways, when going through proof
step by step, you look at a line maybe two or three times
sometimes and can't quite see how it follows from the
previuos one, then suddently do see it, and that is
kind of a bit mysterious too, how that happens.

This has often been written about by mathematicians.

However, it isn't 100% reliable. Sometimes you get a
wonderful idea in much the same fashion, but when you
try to follow it through, it just doesn't work at all.
So one can also make mistakes at this intuitive level
just as one does in the ordinary way of maths.

That has happened to me too. Doesn't seem to get quite
such a write up as the cases where it does work :-)

Robert